As per Wikipedia:
Mantle is a low-level rendering API targeted at 3D video games. Mantle was originally developed by AMD in cooperation with DICE starting in 2013. Mantle was designed as an alternative to Direct3D and OpenGL, primarily for use on personal computers, although suitable hardware is fully available in the Wii U, PlayStation 4 and the Xbox One.
The main advantage of Mantle is the low CPU overhead, which improves performance in games that support it.
The main disadvantage of Mantle is the fact that it's new and games might be buggier in Mantle than in DirectX.
Additionally, since the release of the Mantle API Microsoft has revealed DirectX 12 which is supposed to provide Mantle-like performance and is also meant to be backwards compatible to DirectX 11 hardware, meaning developers may have less reason to use Mantle.